Strong magnetic fields, Dirichlet boundaries, and spectral gaps

Abstract

We consider magnetic Schrödinger operators

$$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$

inL ₂(R ⁿ), where$\vec a \in C^1 (R^n ;R^n )$ and λεR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with$\vec a$, and$M_{\vec a} = \{ x;\vec a(x) = 0\}$, we prove that$H(\lambda \vec a)$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set$M\backslash M_{\vec a}$ has measure zero.

In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that$M\backslash M_{\vec a}$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of$H(\lambda \vec a)$.

We finally address the question of absolute continuity of$\vec a$ for periodic$H(\vec a)$.